The description of multi-phase flows is a primal topic in modern continuum fluid dynamics with enormous applications in biology, chemistry and engineering. Two major approaches have been developed according to the representation of the interface separating the different components (see [1]}, [2]}, [3]}, [4]} and the references therein): Sharp Interface (SI) versus Diffuse Interface (DI) methods (see Figure REF ). In the former, the interface is represented by a hypersurface in the surrounding domain, which leads to the formulation of free boundary problems. In the latter, also called phase-field method, the interface is represented by a layer with finite thickness, corresponding to the level sets of the concentration function whose evolution is governed by a macroscopic equation. The main advantage of this approach is the Eulerian formulation of the phase-field equation, which avoids tracking the interface as in free boundary problems. Although conceptually opposites, these approaches are strictly related since a proper scaling of DI systems approximates SI models in the so-called sharp interface limit (see, e.g., [2]}).
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